Noise-induced contraction of MPO truncation errors in noisy random circuits and Lindbladian dynamics

Abstract

We study how matrix-product-operator (MPO) truncation errors evolve when simulating two setups: (1) 1D Haar-random circuits under either depolarizing noise or amplitude-damping noise, and (2) 1D Lindbladian dynamics of a non-integrable quantum Ising model under either depolarizing or amplitude-damping noise. We first show that the average purity of the system density matrix relaxes to a steady value on a timescale that scales inversely with the noise rate. We then show that truncation errors contract exponentially in both system size $N$ and the evolution time $t$, as the noisy dynamics maps different density matrices toward the same steady state. This yields an empirical bound on the $L_1$ truncation error that is exponentially tighter in $N$ than the existing bound. Together, these results provide empirical evidence that MPO simulation algorithms may efficiently sample from the output of 1D noisy random circuits [setup (1)] at arbitrary circuit depth, and from the steady state of 1D Lindbladian dynamics [setup (2)].

Figures (12)

• Figure 1: Setups. (a) The 1D brickwall circuit of $N$ qubits, where the gates are denoted by blue boxes. After each gate layer, single-qubit noise (depolarizing or amplitude-damping, denoted as red circles) is applied. The dashed boxes illustrate the content and location of the circuit layer ${\cal C}_t$ ($t=2$ illustrated) and the noise layer ${\cal N}_t$ ($t=3$ illustrated). The exact circuit evolution produces a density matrix $\rho_t$. The MPO algorithm simulates the same circuit dynamics, with a truncation followed by normalization (denoted together as ${\cal T_R})$ applied after the corresponding noise layer (location illustrated as the green dashed line for $t=4$), resulting in an approximate density matrix $\sigma_t$. (b) Illustration of 1D Trotterized Lindbladian dynamics of $N$ qubits (hollow circles), consisting of evolution under a local Hamiltonian $H$ (curvy blue arrows) and a dissipator $\cal D$ describing single-qubit noise (depolarizing or amplitude-damping, denoted as wavy black arrows). (c) The corresponding MPO algorithm simulates the dynamics in panels (a,b) approximately, producing an approximation $\sigma_t$ to the density matrix $\rho_t$. We are interested in how the distance between $\rho_t$ and $\sigma_t$ evolves over time.
• Figure 2: Evolution of the $L_2$ norm in noisy circuits, with system size $N=4,6,8,10,12$ (from lighter to darker colors). The circuit depth $t$ takes integer values. (a) Noisy circuits with depolarizing noise for various system sizes $N$ and noise rates $p_{\rm dep}$. The inset shows the extracted early-stage decay rate $\gamma_p$ [cf. \ref{['L2_scale_eq']}] as a function of $p_{\rm dep}$. The fitted line is $\gamma_p = c_{\rm dep} p_{\rm dep}$ with $c_{\rm dep} \approx 2.37$. (b) Same as panel (a) for amplitude-damping noise with rate $p_{\rm damp}$. In addition to the inset showing $\gamma_p$ and its fitted line $\gamma_p = c_{\rm damp} p_{\rm damp}$ with $c_{\rm damp}\approx 1.14$, we also plot, in a separate inset, $\lambda_p$, which characterizes the steady-state purity [cf. \ref{['L2_scale_eq']}] as a function of $p_{\rm damp}$.
• Figure 3: Scaling with system size $N$ of the ratio between the $L_2$ error and the $L_2$ norm, $\|\rho_t-\sigma_t\|_2/\|\rho_t\|_2$, for noisy random-circuit evolution at circuit depth $t=2$ (shown as the square of this ratio). The solid line denotes a linear fit with zero intercept.
• Figure 4: Behavior of $L_2$ truncation errors for noisy random circuits, for (a) depolarizing noise and (b) amplitude-damping noise. The circuit depth $t$ takes integer values. The blue circles denote the single-step error $\| \rho_t - \sigma_{t,T_e=t}\|_2$ [cf. \ref{['L2_step_evo']}], and the blue dashed line denotes the empirical bound on the single-step truncation error \ref{['trunc_error_tr']}. The green markers (with multiple contrast levels) denote the evolution of the single-step error $\| \rho_t - \sigma_{t,T_e\leq t}\|_2$ for several single-time truncation locations $T_e=1,24,30,36$. The red squares denote the total truncation error $\| \rho_t - \sigma_{t}\|_2$, and the red dashed line denotes the predicted empirical bound \ref{['l2_scale_allT']}. The vertical gray dashed lines indicate the depths $T_s$ (where the system almost reaches the steady state) and $2T_s$.
• Figure 5: Scaling behavior of the factor $\Lambda_t$ [\ref{['lambda_t_defi']}] for noisy random circuits, which controls the $L_1$ error $\|\rho_t-\sigma_t\|_1$ via \ref{['l1_scale_allT']}, for the case of (a) depolarizing noise and (b) amplitude-damping noise. The system size is $N=4,6,8,10,12$ (from lighter to darker colors). The circuit depth $t$ takes integer values.